A Second-Order Solution of Saint-Venant's Problem for a Piezoelectric Circular Bar Using Signorini's Perturbation Method

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A Second-Order Solution of Saint-Venant’s Problem for

a Piezoelectric Circular Bar Using Signorini’s

Perturbation Method

Romesh C. Batra, Francesco Dell’Isola, Stefano Vidoli

To cite this version:

A Second-Order Solution of Saint-Venant’s

Problem for a Piezoelectric Circular Bar Using

Signorini’s Perturbation Method

The paper is dedicated with deep respect to Professor Roger Fosdick on his

60th birthday.

R.C. BATRA1_{, F. DELL’ISOLA}2_{and S. VIDOLI}2

1_{Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and} State University, Blacksburg, VA 24061, U.S.A. E-mail: [email protected]

2_{Dipartimento di Ingegneria Strutturale e Geotecnica, Universitá di Roma ‘La Sapienza’, 00184} Roma, Italy

Abstract. We study electromechanical deformations of a homogeneous transversely isotropic piezo-electric prismatic circular bar loaded only at the end faces. The constitutive relations for the material of the bar are taken to be quadratic in the displacement gradients and the electric f eld. It is found that the two end faces of the bar when twisted with no electric charge applied to them will exhibit a difference in the electric potential. Thus the piezoelectric cylinder could be used to measure the torque or the angular twist.

Mathematics Subjcet Classifi ation (1991): 73C99, 73B99. Key words: Poynting effect, non-linear piezoelasticity, twist sensors.

1. Introduction

analyze here electromechanical deformations of a circular cylindrical piezoelectric bar made of a transversely isotropic material. Second order constitutive relations for a piezoelectric material have been derived by Yang and Batra [12]. It is found that the second-order Poisson effect is not of the Saint-Venant type, and even when the bar is deformed by applying pure torques and no electric charges at the end faces, the potential difference between the end faces is proportional to the square of the angular twist.

We note that Batra and Yang [13] have proved Toupin’s version [14] of the Saint-Venant principle for a linear piezoelectric bar. Iesan [15–18] has studied the Saint-Venant problem for inhomogeneous and anisotropic linear elastic bodies, elastic dielectrics, and microstretch elastic solids. Dell’Isola and Rosa [19, 20] and Davi [21] have analyzed the Saint-Venant problem for linear piezoelectric bodies, and dell’Isola and Batra [22] for linear elastic porous solids.

2. Formulation of the Problem

Equations governing quasistatic deformations of a homogeneous transversely iso-tropic piezoelectric body  are

Div(T + TE

)= 0, in , (1.1)

(T + TE)FT = F(T + TE)T, in , (1.2)

Div(D) = 0, in , (1.3)

where T is the f rst Piola–Kirchhoff stress tensor, TE the f rst Piola–Kirchhoff–

Maxwell stress tensor, D the referential electric displacement, and Div is the

di-vergence operator with respect to coordinates in the reference conf guration. These quantities are related to their counterparts in the present conf guration as follows.

T = J σF−1T

, TE = J σEF−1T, D = J F

−1_{D. (2)}

Here J = det F, F is the deformation gradient, σ the Cauchy stress tensor, σE

the Cauchy–Maxwell stress tensor, and D the electric displacement in the present conf guration. Equations (1.1), (1.2) and (1.3) express, respectively, the balance of linear momentum, the balance of moment of momentum, and the Maxwell law for the electric displacement with the body charge density set equal to zero. Constitutive relations for T and TE will be chosen so that (1.2) is identically

satisf ed.

For a piezoelectric material, we introduce, in the present conf guration, electric fi ld ˆE and electric polarization P through

Following Abraham, Einstein and Laub (see [11] Equation 3.6.22,23) we choose the following constitutive equation for σE

σE = Sym(P ⊗ ˆE) + ˆE ⊗ ˆE −1₂ ˆE21, (4)

where Sym(a ⊗ b) = (a ⊗ b + b ⊗ a)/2, 1 is the identity tensor, and the tensor product ⊗ between two vectors a and b is def ned by

(a ⊗ b)c = (b · c)a (5)

for every vector c. Quantities P and ˆE are related to their counterparts 5 and W in the reference conf guration as

5= J F−1P, W = FT ˆE. (6)

Let ψ denote an electric potential f eld in the reference conf guration so that

W = −Grad ψ, (7)

where Grad is the gradient operator in the reference conf guration. The existence of ψ is guaranteed by the referential Maxwell equation Curl W = 0.

We consider a prismatic body occupying the domain  = A × [0, `] in the stress and polarization free reference conf guration with its axis aligned along the direction e of its transverse isotropy. Thus A is the cross-section and ` the length of the body. The mantle of the prismatic body is taken to be free of surface tractions and electric charge, the centroid of the end face A0:= A × {0} is rigidly clamped in the sense that displacements u = x − X, inf nitesimal rotations (H − HT_)/2

and the electric potential ψ there vanish, and surface tractions and electric charge are prescribed on the end faces A0 and A` := A × {`} such that the body is in

equilibrium. Thus

(T + TE)N = 0, D · N = 0 on ∂A × [0, `], (8.1)

(T + TE)e = f, D · e = q on A0 and A`. (8.2)

Here N is an outward unit normal on the mantle ∂A×[0, `], f the prescribed surface traction, q the specif ed electric charge, H = Grad u, x and X denote, respectively, the position of a material point in the present and reference conf gurations. With the origin at the centroid of the cross-section A0, we set

X = r + ze, u = we + v, W = −(ψ0_{e + grad ψ), (9)}

where a prime denotes differentiation with respect to the axial coordinate z. Thus

wand v equal the axial and in-plane components of the displacement u of a point.

and div signify respectively the two-dimensional gradient and divergence operators in the plane A. The integrability conditions for the problem are

Z A f dA ₀ = 0, Z A qdA ₀ = 0, Z Ax ∧ f dA ₀ + x|0 r=0∧ Z Af dA = 0, (10) where a ∧ b = (a ⊗ b − b ⊗ a) for arbitrary vectors a and b. Equations (10) imply that the resultant force and the resultant charge on every cross-section is the same and every portion of the bar is in equilibrium.

3. Signorini’s Expansion

In Signorini’s method, we assume that the displacement u and the electric potential

ψhave a series expansion

u = η˙u + η2_{¨u + . . . , ψ= η ˙ψ + η}2_{¨ψ + . . . , (11)} where η is a small, yet to be determined, parameter in the problem. Surface trac-tions f and the surface charge q are similarly expanded as a power series in η. For a second-order piezoelectric material with null stresses and polarization in the reference conf guration,

T = η˙S + η2_{( ¨S + ˙H˙S), (12.1)}

5= η ˙5 + η2¨5. (12.2)

Here S is the second Piola–Kirchhoff stress tensor, ˙S and ˙5 are homogeneous linear forms in ˙H and ˙W, and ¨S and ¨5 are homogeneous quadratic forms in ˙H and

˙

Div( ¨T + ¨TE_{)= 0, in ,} Div( ¨5 + ¨W + ˙J ˙W − 2(Sym ˙H) ˙W) = 0, in , ( ¨T + ¨TE)N = 0, ( ¨5 + ¨W + ˙J ˙W − 2(Sym ˙H) ˙W) · N = 0, on ∂A × [0, `], ( ¨T + ¨TE<sub>)</sub>e = ¨f, ( ¨5 + ¨W + ˙J ˙W − 2(Sym ˙H) ˙W) · e = ¨q, on A 0 and A`. (14)

In an attempt to express the left-hand sides of Equations (14) for ¨u and ¨ψ in the same form as those of (13) for ˙u and ˙ψ, we decompose additively ¨T and ¨5 as

¨T = ¨¯T + ¨Ts, ¨5 = ¨¯5 + ¨5s. (15)

¨¯T and ¨¯5 are related to ¨u and ¨ψ in the same way as ˙T and ˙5 are to ˙u and ˙ψ, the relation between the former set of variables is given below.

¨¯T = 2µSym grad ¨v + [(c3+ λ) ¨w0_{+ λ div ¨v − e2¨ψ}0_]ˆI +Sym{[ ˜µ(¨v0_{+ grad ¨w) − e3grad ¨ψ] ⊗ e}} +[2(c1+1₂λ+ c3+ c4+ µ) ¨w0+ (c3+ λ) div ¨v

− (e1+ e2+ 2e3) ¨ψ0]e ⊗ e, (16.1)

¨¯5 = 2ε2grad ¨ψ − e3(¨v0+ grad ¨w)

+[2(ε1+ ε2) ¨ψ0− (e1+ e2+ 2e3)¨w0− e2div ¨v]e. (16.2)

Here c1, c3, c4, e1, e2, e3, ε1and ε2are material constants, ˜µ = (c4+2µ)/2, and ˆI is the two-dimensional identity operator. Equations (16.1) and (16.2) are constitutive relations for a linear transversely isotropic piezoelectric material. We presume that the piezoelastic constants λ, µ, c1, c3, c4, e1, e2, e3, ε1and ε2are such that the strain energy density is positive def nite so that the solution of a traction boundary value problem for a linear piezoelectric body is unique to within a rigid body motion. Substitution from (15) into (14) and the integrability conditions (10) yields

Z AX ∧ ( ¨¯Te) dA = Z AX ∧ ¨f dA + R Ms, (17) Z A ( ¨¯5+ ¨W) · e dA = Z A ¨q dA + RQs, Z A ¨¯T0 e dA = hs, Z A (X ∧ ¨¯Te)0dA + e ∧ Z A ¨¯TedA = gs, Z A ( ¨¯5+ ¨W)0· e dA = is, where

bs = −Div( ¨Ts + ¨TE), cs = −Div( ¨5s+ ˙J ˙W − 2(Sym ˙H) ˙W),

fms = −( ¨Ts+ ¨TE)N, qms = −( ¨5s + ˙J ˙W − 2(Sym ˙H) ˙W) · N, RF s = − Z A ( ¨Ts+ ¨TE)e dA, RMs= − Z AX ∧ ( ¨T s + ¨TE)e dA, RQs = − Z A ( ¨5s + ˙J ˙W − 2(Sym ˙H) ˙W) · e dA, hs = − Z A ( ¨Ts+ ¨TE)0e dA, gs = − Z A [X ∧ ( ¨Ts+ ¨TE)e + ˙u ∧ ˙Te]0dA −e ∧ Z ( ¨Ts + ¨TE)e dA − ˙u0|r=0∧ Z A ˙Te dA, is = − Z A ( ¨5s + ˙J ˙W − 2(Sym ˙H) ˙W)0· e dA. (18)

We assume that the bar is initially twisted by an inf nitesimal amount τ and carries a small electric f eld (−ω)e. Its deformations are given by

˙u = −νaωr + zτ(∗r) + zaωe, ˙ψ = zω, (19.1)

where

a = e2

c3, ν=

λ

2(λ + µ), ∗r = e × r. (19.2)

either aω/R or τ. Terms bs, cs, qms, fms, RF s, Rms and RQs in (17) and (18) are

homogeneous quadratic forms in ω and τ and are given below. bs = χ1τ2r + χ2τ2ze, cs = χ3τ2z, fms = (χ4τ2r2+ χ5ω2+ χ6τ2z2)N + χ7τ2(r ⊗ r)N +(χ8τ2zr · N + χ₉τ ω(∗r) · N)e, qms = χ10τ2zr · N + χ11τ ω(∗r) · N, RF s = (χ12τ2JA+ (χ13τ2z2+ χ14ω2)A)e, RMs = χ15τ ωJA(e1∧ e2), RQs = χ16τ2JA+ (χ17τ2z2+ χ18ω2)A. (20)

Expressions for χ1, χ2. . . χ18 in terms of the elastic constants used in the consti-tutive relation are given in the Appendix. A equals the area of cross-section of the bar, JAis the polar moment of inertia, and e1and e2are two orthonormal vectors in A.

Substitution for ¨¯T and ¨¯5 from (16.1) and (16.2) into (17), and recalling (9), we arrive at the following f eld equations for the determination of ¨u and ¨ψ.

F (¨v) + (c3+ λ + ˜µ)grad ¨w0− (e2+ e3)grad ¨ψ0+ ˜µ¨v0= bsA, in A,

1R¨˜w + (c3+ λ + ˜µ)div ¨v0+ 2(c1+ λ₂+ c3+ c4+ µ) ¨w00

− (e1+ e2+ 2e3) ¨ψ00= bse,in A, 1R¨˜ψ − (e2+ e3)div ¨v0+ 2(ε1+ ε2− 1/2) ¨ψ00

− (e1+ e2+ 2e3)¨w00= cs, in A,

G(¨v)N + [(c3+ λ) ¨w0− e2¨ψ0]N = fmsA, on ∂A, grad ¨˜w · N + ˜µ¨v0_{· N = fmse, on ∂A,}

grad ¨˜ψ · N − e3¨v0_{· N = q}

ms, on ∂A,

(21)

where

F (v) = µ1Rv + (λ + µ)grad div v, G(v) = 2µ Sym grad v + λ(div v)ˆI,

¨˜w = ˜µ ¨w − e3¨ψ, ¨˜ψ = −e3¨w + (2ε2− 1) ¨ψ, bs = bsA+ bsee, fms = fmsA+ fmsee,

(22)

A Saint-Venant/Almansi Solution

We seek a solution of (21) of the form ¨w = m X i=0 zi i! ¨wi(r), ¨v = m X i=0 zi i!¨vi(r), ¨ψ = m X i=0 zi i! ¨ψi(r). (23)

Substituting from (23) into (21), recalling (20), and equating like powers of zi_{/ i!}

on both sides, we obtain partial differential equations, boundary conditions and integrability conditions to determine ¨w0, ¨w1, . . . ,¨v0,¨v1, . . . , ¨ψ0, ¨ψ1, . . .. For i >

3, these boundary value problems have null solutions. Denoting constants by a superscript zero, for i = 3, the solution is

¨v3= v0

3+ θ30(∗r), ¨w3= w30, ¨ψ3= ψ30. (24)

The integrability conditions for the torque, axial force and the charge require that

θ₃0= 0, w₃0= 0, ψ₃0= 0. (25)

Using (24) and (25), equations for the determination of ¨v2, ¨w2and ¨ψ2are F (¨v2)= 0, 1R¨˜w2= 0, 1R¨˜ψ₂= 0,

G(¨v2)N = −2(λ + µ)τ2N, grad ¨˜w2· N = − ˜µv0₃· N,

grad ¨˜ψ2· N = e3v03· N

(26) and their solution is

¨v2= v02+ θ20(∗r) − τ2r, ¨w2= w20− v03· r, ¨ψ2= ψ20. (27) The integrability conditions for the torque, axial force and the electric charge re-quire that

θ₂0= 0, w₂0= 0, ψ₂0= 0. (28)

Field equations for ¨v1, ¨w1and ¨ψ1are

F (¨v1)= (c3+ λ)v0₃, 1R¨˜w1= 0, 1R¨˜ψ₁= 0,

G(¨v1)N = (c₃+ λ)(v0₃· r)N, grad ¨˜w1· N = − ˜µv0₂· N,

grad ¨ψ1· N = e3v0 2· N,

(29) and have the solution

Equations for f nding f elds ¨v0, ¨w0and ¨ψ0can now be written as F (¨v0)= (c3+ λ)v0₂+ ( ˜µ + χ1)τ2r, 1R˜w0= ξ1v0₃· r, 1R˜ψ0= ξ3v0₃· r, G(¨v0)N = [(c₃+ λ)(v0₂· r) + (e2ψ₁0− (c3+ λ)w0₁) + (χ4τ2r2+ χ5ω2)]N + χ7τ2(r ⊗ r)N, (grad ˜w0)· N = − ˜µv0₁· N + ξ2[Sym(r ⊗ (∗r))(∗v0₃)] · N, (grad ˜ψ0)· N = e3v0₁· N + +ξ4[Sym(r ⊗ (∗r))(∗v0₃)] · N, (31)

where expressions for ξ1, ξ2, ξ3, ξ4and other ξ’s introduced below in terms of other material parameters are given in the Appendix. The solution of (31) is

¨v0= v0 0+ θ00(∗r) + (c3+ λ) 2(λ + µ)Sym(r ⊗ (∗r))(∗v02) +(χ5ω2+ ξ0R2τ2+ e2ψ10− (c3+ λ)w01) 2(λ + µ) r + ξ5τ2r2r, ¨w0= w00− v01· r + ξ68+ ξ79, ¨ψ_{0= ψ}0 0+ ξ88+ ξ99, (32)

where functions 8 and 9 are given by

8= 1₈[(4ξ2− 3ξ1)R2+ ξ1r2]v0₃· r,

9= 1₈[(4ξ4− 3ξ3)R2+ ξ3r2]v0₃· r. (33)

The clamping conditions u = 0, H − HT _{= 0, ψ = 0 at the centroid of A0}

require that v0

0= 0, w00= 0, θ00= 0, ψ00= 0, v01= 0. (34)

The second-order solution is characterized by seven constants v0

3, v02, θ10, w10 and

ψ₁0 representing second-order f exure, bending, torsion, elongation and electric

potential respectively. However, these effects are coupled in the sense that if a piezoelectric circular bar is twisted by applying equal and opposite torques at the end faces, then there is also second-order torsion, elongation and electric f eld.

Here T is the torque and Q the total charge. The surface tractions f at the end faces A0 and A` have zero resultant force, and their resultant moment equals T about

the axis e of the bar. The solution of the second-order problem is

v = −νarω + z(∗r)τ + [ξ12ω2+ (ξ13R2+ ξ5r2)τ2]r + χ15_˜µ z(∗r)τω − 1₂z2rτ2, w= zaω + ξ10[(2ξ5(λ+ µ) + ξ0)R2τ2+ χ5ω2]z,

ψ= zω + ξ11[(2ξ5(λ+ µ) + ξ0)R2τ2+ χ5ω2]z, ω= Q/ξ14A, τ = T / ˜µJA

(35) Thus the angle of twist/length equals τ + (χ15/˜µ)τω implying thereby that an

electric f eld alters the angle of twist/length and this change is proportional to the charge/area. Also there is a second-order Poisson effect with one part proportional to r and another one proportional to r3_{; the part varying as r}3 _{depends upon the} piezoelectric constants.

One part of the axial strain w0 _{is proportional to τ}2 _{and ω}2 _{as expected and} is a generalization of the Poynting effect to transversely isotropic piezoelectric materials. When τ = 0, the term χ5ξ10ω2 represents the correction to the axial

strain caused by the nonlinear response of the piezoelectric cylinder to the applied electric f eld.

Equation (35)3 indicates that the difference of the electric potential at the two end faces of the piezoelectric cylinder depends upon the square of the angular twist. Even when there is no charge applied at the end faces, twisting of the piezoelectric cylinder will induce a measurable difference in the electric potential between the end faces. Hence a piezoelectric cylinder can be used to measure the angular twist. 4. Conclusions

We have studied the electromechanical deformations of a second-order, transversely isotropic homogeneous circular cylindrical bar with mechanical loads and/or elec-tric charges applied to its end faces only. The constitutive relations are taken to be quadratic in the displacement gradients and the electric f eld. The centroid of one end cross-section is rigidly clamped in the sense that displacements, inf nitesimal rotations and the electric potential vanish there.

Appendix

Using the notations

˙E = ( ˙H + ˙H)T_/2, ¨E = ˙HT ˙H/2,

˙I₁= e · ( ˙Ee), ¨I1= e · ( ¨Ee), ˙I2= tr ˙H, ¨I2= tr ¨E, ˙I3= ˙W · e, ¨51= e · ( ˙E2e), ¨52= (tr ˙E)2,

¨5₃= ˙W · ˙W, ¨5₄= e · ( ˙E ˙W) + ˙W · ( ˙Ee), ¨I3= ¨W · e,

we f nd that the constitutive relations for a second-order transversely isotropic material with the axis of transverse isotropy along the unit vector e are as follows:

˙S = (2c1˙I_{1+ c3}˙I_{2+ e1}˙I₃)e ⊗ e + (2c2˙I2+ c3˙I1+ e2˙I3)1

+c4Sym(e ⊗ ˙Ee) + 2c5˙E + e3Sym(e ⊗ ˙W), ¨S = [2c1¨I_{1+ c3}¨I_{2+ 3λ1}˙I2

1 + 2λ3˙I1˙I2+ λ4˙I22+ λ5¨51

+λ7¨52+ 2ν1˙I1˙I3+ ν2˙I32+ ν7¨53+ ν9¨54+ ν14˙I2˙I3]e ⊗ e +[2c2¨I2+ c3¨I1+ 3λ2˙I22+ λ3˙I12+ 2λ4˙I1˙I2+ λ6¨51+ λ8¨52

+ 2ν3˙I2˙I3+ ν4˙I32+ ν8¨53+ ν10 ¨54+ ν14˙I1˙I3]1

+2c4Sym(e ⊗ ˙Ee) + 2(λ5˙I1+ λ6˙I2+ ν5˙I3)Sym(e ⊗ ˙Ee) +2c5¨E + 2(λ7˙I1+ λ8˙I2+ ν6˙I3) ˙E

+2(ν9˙I1+ ν10˙I2+ ν11˙I3)Sym(e ⊗ ˙W)

+3λ9( ˙E)2+ ν12W ⊗ ˙W + 2ν˙ 13Sym(e ⊗ ˙E ˙W + ˙W ⊗ ˙Ee), ˙5 = −(2ε1˙I₃+ e1˙I1+ e2˙I2)e − 2ε2W − 2e˙ 3˙Ee,

¨5 = −[e1¨I1+ e2¨I2+ 3µ1˙I32+ µ2¨53+ ν1˙I12+ 2ν2˙I3˙I1+ ν3˙I22+ 2ε1¨I3 + 2ν4˙I3˙I2+ ν5¨51+ ν6¨52+ ν11¨54+ ν14˙I1˙I2]e

−2[µ2˙I3+ ν7˙I1+ ν8˙I2] ˙W − 2e3¨Ee − 2ε2W¨ −2(ν9˙I1+ ν10˙I2+ ν11˙I3) ˙Ee − 2ν12˙E ˙W − 2ν13¨E2e.

Here c1, c2, . . . , e1, e2, . . . , λ1, λ2, . . . , ν1, ν2, . . . , ε1, ε2, . . ., and µ1, µ2, . . .are

material parameters. Expressions for other material parameters used in the text are given below.

+16λλ6µ+ 24λλ7µ+ 16λλ8µ +24λλ9µ+ 12c4µ2+ 30λµ2+ 12λ1µ2+ 12λ2µ2 +12λ3µ2+ 12λ4µ2+ 12λ5µ2+ 12λ6µ2 +12λ7µ2+ 12λ8µ2+ 12λ9µ2+ 12µ3 +12c1(λ+ µ)2)+ 2c32(λ+ µ)2 ×(−1 + 4ε1+ 4ε2− 4ν11− 2ν12− 2ν2− 2ν4− 2ν7− 2ν8) +c3e₂(−e2(5λ2+ 16λµ + 12µ2) +4(λ + µ)(2λν1+ 2µν1+ 2λν11+ 4µν10+ 4λν13 +4µν13+ λν14+ 2µν14+ 2µν3+ 2λν5+ 2µν5 +2λν6+ 2µν6+ 4λν9+ 4µν9))), χ₁₅= 1 4c3(λ+ µ)(−e2(6c4λ+ 2λλ5+ 4λλ7 +3λλ9+ 8c4µ+ 8λµ + 2λ5µ+ 2λ6µ +4λ7µ+ 4λ8µ+ 6λ9µ+ 8µ2+ 8c1(λ+ µ))

+2c3(3e3λ− 2e2µ+ 3e3µ+ 2e1(λ+ µ)

+2λν13+ 2µν13+ λν5+ µν5+ 2λν6+ 2µν6)), χ16= 1₂e1+1₂e2+ e3+1₂ν13+1₄ν5+1₂ν6, χ17= e2, χ18= 1 4c2 3(λ+ µ)2 (12c32(λ+ µ)2(µ1+ µ2) −4c3e2(λ+ µ)(λ(2 + 3ν11+ 2ν12+ 2ν2+ 2ν7) +µ(1 + 3ν11+ 2ν12+ 2ν2+ 2ν4+ 2ν7+ 2ν8))

+e22_(3e2λ2_{+ 4e3λ}2_{+ 4e2λµ+ 8e3λµ} +2e2µ2+ 4e3µ2+ 2e1(λ+ µ)2

χ₁₉= −2(c3+ λ), χ20= 2e2, ξ0= χ4+ χ7−_{4(λ + 2µ)}2λ + 3µ , ξ1= 2c1+1₂λ+ c3+ c4+ µ − c3+ λ +1₂c4+ µ  c3+ λ λ+ µ  , ξ₂= −(1₂c₄+ µ) (c3+ λ) (2λ + 2µ), ξ3= (e2+ e3)(c3+ λ) (λ+ µ) − (e1+ e2+ 2e3), ξ4= e3(c3+ λ) (2λ + 2µ), ξ₅= −4c3+ 2c4− 4λ − 2λ6− 4λ8+ 3λ9+ 4µ 32(λ + 2µ) , ξ6= 1 − 2ε2 e₃2+ ˜µ − 2ε2˜µ, ξ₇= − e3 e₃2+ ˜µ − 2ε2˜µ, ξ₈= − e3 e₃2+ ˜µ − 2ε2˜µ, ξ9= − ˜µ e₃2+ ˜µ − 2ε2˜µ, ξ₁₀ = (−c3− e1e₂− e22− 2e2e₃+ 2c3ε₁ +2c3ε₂− λ + 2ε1λ+ 2ε2λ)/(2c4e₂2 +c32_{(−1 + 21+ 22)+ 2c4λ}

+e12λ+ 4e1e3λ+ 4e32λ− 4c4ε1λ− 4c4ε2λ

+2c4µ+ e12µ+ 2e1e₂µ+ 3e22µ

+4e1e3µ+ 4e2e3µ+ 4e32µ− 4c4ε1µ

−4c4ε2µ+ 3λµ − 6ε1λµ− 6ε2λµ+ 2µ2− 4ε1µ2− 4ε2µ2

+2c1(e22+ λ − 2ε1λ− 2ε2λ+ µ

ξ₁₁ = (−2c1e₂− 2c4e₂+ c3(e₁− e2+ 2e3)+ e1λ+ 2e3λ− 2e2µ)/ × (λ+ µ)  (e1+ e2+ 2e3)−e2(c3+ λ) λ+ µ 2 −  −1 + 2ε1+ 2ε2− e22 λ+ µ  ×  2(c1+ c3+ c4+1₂λ+ µ) − (c3+ λ) 2 λ+ µ  , ξ₁₂ = χ5(1 + e2ξ11− (c3+ λ)ξ10) 2(λ + µ) , ξ13 = ξ5(e2ξ11− (c3+ λ)ξ10)+ e2ξ0ξ10 2(λ + µ) − (c₃+ λ)ξ0ξ₁₁ 2(λ + µ) , ξ14 = (2ε1+ 2ε2− 1)c3(λ+ µ) − e2(e2µ+ e1(λ+ µ) + 2e3(λ+ µ)) c3(λ+ µ) . Acknowledgements

R.C. Batra’s work was partially supported by the NSF Grant CMS9713453 and the ARO grant DAAG55-98-1-0030 to Virginia Polytechnic Institute and State Uni-versity. S. Vidoli and F. dell’Isola thank Prof. N. Rizzi for having secured f nancial support for their visits to Virginia Tech. Many of the algebraic operations were performed by using Mathematica.

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